Metamath Proof Explorer


Theorem weiunfr

Description: A well-founded relation on an indexed union can be constructed from a well-ordering on its index class and a collection of well-founded relations on its members. (Contributed by Matthew House, 23-Aug-2025)

Ref Expression
Hypotheses weiun.1 F = w x A B ι u x A | w B | v x A | w B ¬ v R u
weiun.2 T = y z | y x A B z x A B F y R F z F y = F z y F y / x S z
Assertion weiunfr R We A R Se A x A S Fr B T Fr x A B

Proof

Step Hyp Ref Expression
1 weiun.1 F = w x A B ι u x A | w B | v x A | w B ¬ v R u
2 weiun.2 T = y z | y x A B z x A B F y R F z F y = F z y F y / x S z
3 csbeq1 s = ι p F r | q F r ¬ q R p s / x S = ι p F r | q F r ¬ q R p / x S
4 csbeq1 s = ι p F r | q F r ¬ q R p s / x B = ι p F r | q F r ¬ q R p / x B
5 3 4 freq12d s = ι p F r | q F r ¬ q R p s / x S Fr s / x B ι p F r | q F r ¬ q R p / x S Fr ι p F r | q F r ¬ q R p / x B
6 simpl3 R We A R Se A x A S Fr B r x A B r x A S Fr B
7 nfv s S Fr B
8 nfcsb1v _ x s / x S
9 nfcsb1v _ x s / x B
10 8 9 nffr x s / x S Fr s / x B
11 csbeq1a x = s S = s / x S
12 csbeq1a x = s B = s / x B
13 11 12 freq12d x = s S Fr B s / x S Fr s / x B
14 7 10 13 cbvralw x A S Fr B s A s / x S Fr s / x B
15 6 14 sylib R We A R Se A x A S Fr B r x A B r s A s / x S Fr s / x B
16 simpl1 R We A R Se A x A S Fr B r x A B r R We A
17 simpl2 R We A R Se A x A S Fr B r x A B r R Se A
18 1 2 16 17 weiunlem2 R We A R Se A x A S Fr B r x A B r F : x A B A t x A B t F t / x B s A t s / x B ¬ s R F t
19 18 simp1d R We A R Se A x A S Fr B r x A B r F : x A B A
20 19 fimassd R We A R Se A x A S Fr B r x A B r F r A
21 eqid ι p F r | q F r ¬ q R p = ι p F r | q F r ¬ q R p
22 simprl R We A R Se A x A S Fr B r x A B r r x A B
23 simprr R We A R Se A x A S Fr B r x A B r r
24 1 2 16 17 21 22 23 weiunfrlem R We A R Se A x A S Fr B r x A B r ι p F r | q F r ¬ q R p F r t r ¬ F t R ι p F r | q F r ¬ q R p t r ι p F r | q F r ¬ q R p / x B F t = ι p F r | q F r ¬ q R p
25 24 simp1d R We A R Se A x A S Fr B r x A B r ι p F r | q F r ¬ q R p F r
26 20 25 sseldd R We A R Se A x A S Fr B r x A B r ι p F r | q F r ¬ q R p A
27 5 15 26 rspcdva R We A R Se A x A S Fr B r x A B r ι p F r | q F r ¬ q R p / x S Fr ι p F r | q F r ¬ q R p / x B
28 inss2 r ι p F r | q F r ¬ q R p / x B ι p F r | q F r ¬ q R p / x B
29 28 a1i R We A R Se A x A S Fr B r x A B r r ι p F r | q F r ¬ q R p / x B ι p F r | q F r ¬ q R p / x B
30 vex r V
31 30 inex1 r ι p F r | q F r ¬ q R p / x B V
32 31 a1i R We A R Se A x A S Fr B r x A B r r ι p F r | q F r ¬ q R p / x B V
33 19 ffund R We A R Se A x A S Fr B r x A B r Fun F
34 fvelima Fun F ι p F r | q F r ¬ q R p F r t r F t = ι p F r | q F r ¬ q R p
35 33 25 34 syl2anc R We A R Se A x A S Fr B r x A B r t r F t = ι p F r | q F r ¬ q R p
36 simprl R We A R Se A x A S Fr B r x A B r t r F t = ι p F r | q F r ¬ q R p t r
37 simplrl R We A R Se A x A S Fr B r x A B r t r F t = ι p F r | q F r ¬ q R p r x A B
38 37 36 sseldd R We A R Se A x A S Fr B r x A B r t r F t = ι p F r | q F r ¬ q R p t x A B
39 18 simp2d R We A R Se A x A S Fr B r x A B r t x A B t F t / x B
40 39 r19.21bi R We A R Se A x A S Fr B r x A B r t x A B t F t / x B
41 38 40 syldan R We A R Se A x A S Fr B r x A B r t r F t = ι p F r | q F r ¬ q R p t F t / x B
42 simprr R We A R Se A x A S Fr B r x A B r t r F t = ι p F r | q F r ¬ q R p F t = ι p F r | q F r ¬ q R p
43 42 csbeq1d R We A R Se A x A S Fr B r x A B r t r F t = ι p F r | q F r ¬ q R p F t / x B = ι p F r | q F r ¬ q R p / x B
44 41 43 eleqtrd R We A R Se A x A S Fr B r x A B r t r F t = ι p F r | q F r ¬ q R p t ι p F r | q F r ¬ q R p / x B
45 36 44 elind R We A R Se A x A S Fr B r x A B r t r F t = ι p F r | q F r ¬ q R p t r ι p F r | q F r ¬ q R p / x B
46 45 ne0d R We A R Se A x A S Fr B r x A B r t r F t = ι p F r | q F r ¬ q R p r ι p F r | q F r ¬ q R p / x B
47 35 46 rexlimddv R We A R Se A x A S Fr B r x A B r r ι p F r | q F r ¬ q R p / x B
48 27 29 32 47 frd R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n
49 simprl R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n n r ι p F r | q F r ¬ q R p / x B
50 49 elin1d R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n n r
51 fveq2 t = o F t = F o
52 51 breq1d t = o F t R ι p F r | q F r ¬ q R p F o R ι p F r | q F r ¬ q R p
53 52 notbid t = o ¬ F t R ι p F r | q F r ¬ q R p ¬ F o R ι p F r | q F r ¬ q R p
54 24 ad2antrr R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r ι p F r | q F r ¬ q R p F r t r ¬ F t R ι p F r | q F r ¬ q R p t r ι p F r | q F r ¬ q R p / x B F t = ι p F r | q F r ¬ q R p
55 54 simp2d R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r t r ¬ F t R ι p F r | q F r ¬ q R p
56 simpr R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r o r
57 53 55 56 rspcdva R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r ¬ F o R ι p F r | q F r ¬ q R p
58 fveqeq2 t = n F t = ι p F r | q F r ¬ q R p F n = ι p F r | q F r ¬ q R p
59 54 simp3d R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r t r ι p F r | q F r ¬ q R p / x B F t = ι p F r | q F r ¬ q R p
60 simplrl R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r n r ι p F r | q F r ¬ q R p / x B
61 58 59 60 rspcdva R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r F n = ι p F r | q F r ¬ q R p
62 61 breq2d R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r F o R F n F o R ι p F r | q F r ¬ q R p
63 57 62 mtbird R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r ¬ F o R F n
64 breq1 m = o m ι p F r | q F r ¬ q R p / x S n o ι p F r | q F r ¬ q R p / x S n
65 64 notbid m = o ¬ m ι p F r | q F r ¬ q R p / x S n ¬ o ι p F r | q F r ¬ q R p / x S n
66 simprr R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n
67 66 ad2antrr R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r F o = F n m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n
68 simplr R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r F o = F n o r
69 id t = o t = o
70 51 csbeq1d t = o F t / x B = F o / x B
71 69 70 eleq12d t = o t F t / x B o F o / x B
72 39 ad3antrrr R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r F o = F n t x A B t F t / x B
73 22 ad2antrr R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r r x A B
74 73 56 sseldd R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r o x A B
75 74 adantr R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r F o = F n o x A B
76 71 72 75 rspcdva R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r F o = F n o F o / x B
77 simpr R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r F o = F n F o = F n
78 61 adantr R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r F o = F n F n = ι p F r | q F r ¬ q R p
79 77 78 eqtrd R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r F o = F n F o = ι p F r | q F r ¬ q R p
80 79 csbeq1d R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r F o = F n F o / x B = ι p F r | q F r ¬ q R p / x B
81 76 80 eleqtrd R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r F o = F n o ι p F r | q F r ¬ q R p / x B
82 68 81 elind R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r F o = F n o r ι p F r | q F r ¬ q R p / x B
83 65 67 82 rspcdva R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r F o = F n ¬ o ι p F r | q F r ¬ q R p / x S n
84 79 csbeq1d R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r F o = F n F o / x S = ι p F r | q F r ¬ q R p / x S
85 84 breqd R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r F o = F n o F o / x S n o ι p F r | q F r ¬ q R p / x S n
86 83 85 mtbird R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r F o = F n ¬ o F o / x S n
87 86 ex R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r F o = F n ¬ o F o / x S n
88 imnan F o = F n ¬ o F o / x S n ¬ F o = F n o F o / x S n
89 87 88 sylib R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r ¬ F o = F n o F o / x S n
90 pm4.56 ¬ F o R F n ¬ F o = F n o F o / x S n ¬ F o R F n F o = F n o F o / x S n
91 90 biimpi ¬ F o R F n ¬ F o = F n o F o / x S n ¬ F o R F n F o = F n o F o / x S n
92 63 89 91 syl2anc R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r ¬ F o R F n F o = F n o F o / x S n
93 92 intnand R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r ¬ o x A B n x A B F o R F n F o = F n o F o / x S n
94 1 2 weiunlem1 o T n o x A B n x A B F o R F n F o = F n o F o / x S n
95 93 94 sylnibr R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r ¬ o T n
96 95 ralrimiva R We A R Se A x A S Fr B r x A B r n r ι p F r | q F r ¬ q R p / x B m r ι p F r | q F r ¬ q R p / x B ¬ m ι p F r | q F r ¬ q R p / x S n o r ¬ o T n
97 48 50 96 reximssdv R We A R Se A x A S Fr B r x A B r n r o r ¬ o T n
98 97 ex R We A R Se A x A S Fr B r x A B r n r o r ¬ o T n
99 98 alrimiv R We A R Se A x A S Fr B r r x A B r n r o r ¬ o T n
100 df-fr T Fr x A B r r x A B r n r o r ¬ o T n
101 99 100 sylibr R We A R Se A x A S Fr B T Fr x A B